Re: properties of integers

where each is a different prime (if a prime does not occur in the factorization of one of m or n, we can take that exponent to be 0),

then

where .

it's easier to explain this with an example:

suppose we have 70 and 90 and we want to find the smallest number that is divisible by both. first, we factor 70 and 90 into primes:

70 = (2)(5)(7)
90 = (2)(3^2)(5)

the primes in this case are 2,3,5 and 7. re-writing both numbers in terms of these primes:

70 = (2^1)(3^0)(5^1)(7^1)
90 = (2^1)(3^2)(5^1)(7^0)

so the least common multiple will be of the form:

(2^a)(3^b)(5^c)(7^d) where:

a = max(1,1), b = max(0,2), c = max(1,1), d = max(1,0).

so a = 1, b = 2, c = 1, d = 1 (all we're doing is picking the "biggest power" of any prime we find in the prime factorizations).

so lcm(70,90) = (2^1)(3^2)(5^1)(7^1) = (2)(9)(5)(7) = 630.

there is also another way of doing this, which is to find the greatest common divisor (factor) of 70 and 90 (which we can do 2 ways, one of which is to pick the "smallest power" of any prime which occurs in both factorizations, or the following way):